nth Term
The nth term of a GP is a_n = a·r^(n−1), where a is the first term and r the common ratio. The exponent is (n−1), not n — the first term carries no factor of r, the second carries r¹, and so on. This single formula lets you jump straight to the 15th or 50th term without writing the whole sequence. The CAT-smart move is to work with ratios of terms: a_m/a_n = r^(m−n), which kills the unknown a. If the 4th term is 24 and the 7th is 192, then a_7/a_4 = r³ = 192/24 = 8, so r = 2 in one line. To check whether three numbers are in GP, test whether the middle term squared equals the product of the outers (b² = ac); this is also how you spot the geometric mean. Watch the sign of r: a negative ratio makes the terms alternate in sign, a trap CAT exploits.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Formula Reference Sheet
Core GP formulas
| nth term | a_n = a·r^(n−1) |
|---|---|
| Sum of n terms (r ≠ 1) | S_n = a(r^n − 1)/(r − 1) |
| Sum of n terms (|r| < 1 form) | S_n = a(1 − r^n)/(1 − r) |
| Infinite sum (|r| < 1) | S_∞ = a/(1 − r) |
| Geometric mean of a and b | GM = √(ab) |
CAT power-tools
| Three terms in GP | a/r, a, ar (product = a³) |
|---|---|
| Ratio of two terms | a_m / a_n = r^(m − n) |
| n GMs between a and b | common ratio r = (b/a)^(1/(n+1)) |
| Sum × (r − 1) | S_n(r − 1) = a(r^n − 1) |
| Each term squared | a², a²r², a²r⁴, ... is a GP with ratio r² |